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Elliptic geometry

Elliptic geometry is a non-Euclidean geometry characterized by positive curvature, where the parallel postulate is replaced by the axiom that no parallel lines exist through a point not on a given line, ensuring all lines intersect.^[1] It models the geometry of a sphere's surface, with points represented as pairs of antipodal points on the sphere and lines as great circles, the shortest paths between points.^[2] Unlike Euclidean geometry, elliptic geometry features a finite, closed space without boundaries, and congruence transformations correspond to rotations of the sphere about its center.^[3] Key properties include the fact that the sum of the interior angles of any triangle exceeds 180 degrees, with the excess proportional to the triangle's area via Girard's theorem, which states that the area equals (sum of angles minus π) times the square of the sphere's radius.^[4] There are no parallel lines, and betweenness is defined through separation axioms rather than the linear ordering of Euclidean geometry, leading to the absence of similar but non-congruent triangles.^[1] Distances are measured along great circle arcs in radian units, and the geometry is conformal under stereographic projection, mapping elliptic lines to circles or lines in the plane.^[2] Elliptic geometry can be formalized through several models, including the spherical model using rotations of the 2-sphere, the projective plane model where antipodal points on the unit disk are identified, and the disk model employing Möbius transformations that preserve antipodal points.^[3] The group of transformations consists of orientation-preserving isometries, such as those given by z \mapsto e^{i\theta} \frac{z - a}{1 - \overline{a}z} in the complex plane, corresponding to rotations via stereographic projection.^[3] This geometry deviates from neutral geometry by altering incidence and order axioms, resulting in properties like the Pythagorean theorem not holding and all perpendiculars to a line intersecting at a single point.^[4]

Basic Concepts

Definitions and axioms

Elliptic geometry is a non-Euclidean geometry defined on the surface of a sphere where antipodal points are identified, treating each pair of opposite points as a single point in the space. This identification results in a finite, closed manifold without boundaries, where the total space has the topology of a projective plane.^[4] In this framework, points correspond to pairs of antipodal points on the sphere, and lines are the great circles passing through these points.^[5] This model, known as single elliptic geometry, is obtained as the quotient of spherical geometry by identifying antipodal points, with spherical geometry serving as a double cover and the identification ensuring unique intersections without redundancy from antipodes.^[5] The foundational axioms of elliptic geometry build upon those of neutral geometry but diverge critically in the treatment of parallels and intersections. A core incidence axiom states that any two distinct lines intersect in exactly one point, eliminating the possibility of parallel lines entirely.^[6] This replaces Playfair's axiom from Euclidean geometry—which posits that through a point not on a given line, exactly one parallel can be drawn—with the elliptic version: through any point not on a given line, every line through that point intersects the given line.^[4] Geodesics in elliptic geometry are the shortest paths along these great circle lines, and all geodesics intersect due to the closed nature of the space.^[5] The elliptic distance between two points is measured as the length of the shorter arc along the connecting great circle, normalized such that the maximum distance is half the circumference of the sphere (π in units where the radius is 1), to avoid measuring the longer arc that would pass through the antipodal identification.^[5] Separation and betweenness axioms ensure that points on a line are ordered uniquely, with segments defined relative to this ordering.^[6] Additionally, the sum of the interior angles in any triangle exceeds π radians (180 degrees), reflecting the positive curvature of the space.^[7]

Relation to spherical geometry

Spherical geometry is the study of geometric figures on the surface of a sphere, where the geodesics, or "lines," are the great circles formed by the intersections of the sphere with planes passing through its center. In this geometry, the space has constant positive curvature, and any two great circles intersect at exactly two antipodal points, leading to a structure where distances are bounded and paths can wrap around the sphere.^[8] Elliptic geometry arises as a quotient of spherical geometry by identifying each pair of antipodal points on the sphere, resulting in a space where every point corresponds to a unique pair of opposite points from the original sphere.^[9] This identification, known as the antipodal quotient, transforms the sphere S^n into the real projective space \mathbb{RP}^n, endowing elliptic geometry with the metric inherited from the sphere but adjusted for the equivalence.^[8] The sphere thus serves as a double cover of the elliptic space, resolving the issue in spherical geometry where geodesics intersect twice by merging those points into single intersection points in the elliptic setting. This quotient construction ensures that elliptic geometry maintains the constant positive curvature of spherical geometry while eliminating the redundancy of antipodal duplicates, such as halving the maximum geodesic distance from \pi r to \pi r / 2, where r is the sphere's radius.^[8] Consequently, the elliptic space is compact, finite, and closed without boundary, contrasting with the infinite extent of the Euclidean plane; its total area or volume is half that of the corresponding sphere, directly determined by the radius r. This finite nature aligns with the modified parallel postulate in elliptic geometry, where no parallel lines exist, as all geodesics intersect.^[9]

Models

Spherical model

The spherical model realizes elliptic geometry by constructing the space as a quotient of the standard n-sphere. Specifically, the n-dimensional elliptic space is defined as the quotient S^n / ~, where S^n denotes the n-sphere embedded in \mathbb{R}^{n+1} equipped with the round metric, and the equivalence relation ~ identifies each point p with its antipodal point -p.^[10] This identification ensures that the model captures the topology of real projective space \mathbb{RP}^n while inheriting the Riemannian structure from the sphere.^[10] The metric on this quotient space is induced by the round metric on S^n, making it a Riemannian manifold of constant positive sectional curvature. For a sphere of radius R, the Gaussian curvature is K = 1/R^2. Points in the model are represented by unit vectors in \mathbb{R}^{n+1}, and the distance between two such points x and y is given by d(x, y) = \arccos(|\langle x, y \rangle|), where \langle \cdot, \cdot \rangle denotes the standard inner product; this formula yields distances in the interval [0, \pi/2] due to the antipodal identification, corresponding to the length of the shortest great circle arc connecting the points or their antipodes.^[4]^[10] Coordinates on the spherical model employ hyperspherical coordinates on S^n, adapted to the quotient by restricting angular ranges to account for antipodal symmetry; for instance, in the two-dimensional case, these reduce to spherical coordinates (\theta, \phi) with \theta \in [0, \pi/2] and \phi \in [0, \pi) to avoid redundancy. The line element in two dimensions, for the unit sphere, is

ds^2 = d\theta^2 + \sin^2 \theta \, d\phi^2,

which generalizes to higher dimensions via the standard expression for the round metric on S^n in hyperspherical coordinates:

ds^2 = d\chi^2 + \sin^2 \chi \left( d\theta_1^2 + \sin^2 \theta_1 \, d\theta_2^2 + \cdots + \sin^2 \theta_1 \cdots \sin^2 \theta_{n-2} \, d\phi^2 \right),

where \chi \in [0, \pi/2] in the elliptic model. This construction provides an intuitive geometric realization, with spherical geometry serving as a double cover of the elliptic space.^[10]

Projective model

The projective model of elliptic geometry identifies elliptic n-space with the real projective space \mathbb{RP}^n, defined as the set of all 1-dimensional linear subspaces (lines through the origin) of \mathbb{R}^{n+1}.^[11] This construction endows the space with a natural projective structure, where the topology and geometry arise from the quotient of the unit sphere S^n \subset \mathbb{R}^{n+1} by antipodal identification, ensuring a compact manifold without boundary.^[12] Points in \mathbb{RP}^n are represented using homogeneous coordinates [x_0 : x_1 : \dots : x_n], where (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} \setminus \{0\} and two vectors define the same point if one is a scalar multiple of the other.^[12] To compute distances, representatives are normalized to unit vectors on the sphere, leveraging the linear algebra of the embedding space. Lines in this model are the projectivizations of 2-dimensional linear subspaces (2-flats) of \mathbb{R}^{n+1}, which intersect the unit sphere in great circles; the projective incidence structure ensures these lines are closed geodesics of length \pi, free from endpoints or infinite extent.^[12]^[11] The distance d between two points [ \mathbf{x} ] and [ \mathbf{y} ], with unit vector representatives \mathbf{x}, \mathbf{y} \in S^n, is defined by

\cos d = | \mathbf{x} \cdot \mathbf{y} |,

so d = \arccos( | \mathbf{x} \cdot \mathbf{y} | ), ranging from 0 to \pi/2.^[13] This formula captures the minimal angular separation on the sphere, accounting for antipodal equivalence, and induces a Riemannian metric of constant sectional curvature 1 on \mathbb{RP}^n.^[12] The metric tensor in homogeneous coordinates derives from the round metric on the sphere, simplified for the real elliptic case by restricting to the orthogonal complement of the radial direction in \mathbb{R}^{n+1}; locally, in affine charts, it takes the form ds^2 = \frac{ 4 \sum dx_i^2 }{ (1 + |x|^2)^2 } after normalization.^[12] This algebraic approach highlights the model's invariance under the action of the projective linear group PGL(n+1, \mathbb{R}), emphasizing incidence over embedded metrics.^[11]

Stereographic model

The stereographic model embeds elliptic geometry conformally into Euclidean space minus a point, enabling practical computations and visualizations by representing elliptic points as coordinates in \mathbb{R}^n \setminus \{0\}. This approach leverages the identification of antipodal points on the unit sphere to model the projective space underlying elliptic geometry.^[14] The construction begins with the standard stereographic projection from the north pole of the unit sphere S^n onto the equatorial hyperplane, which maps points (X_1, \dots, X_n, Z) on the sphere (with Z < 1) to (x_1, \dots, x_n) \in \mathbb{R}^n via x_i = X_i / (1 - Z). To obtain the elliptic model, antipodal points on the sphere are quotiented, effectively identifying points x and -1/|x|^2 \cdot x in the plane (inversion through the unit sphere), yielding a conformal representation of the elliptic space \mathbb{RP}^n minus a point. This quotient ensures that the entire elliptic space is covered without redundancy, with the origin corresponding to the south pole and the projection avoiding the north pole.^[15]^[11] In two dimensions, the model uses complex coordinates z = x + iy on the plane, where a sphere point (x, y, w) with w < 1 maps to z = (x + iy)/(1 - w), and the elliptic identification pairs z with -1/\bar{z}. This adaptation allows elliptic lines—great circles on the sphere—to appear as circles or lines in the plane that pass through the origin or are orthogonal to the unit circle.^[16]^[11] The induced metric is conformal to the Euclidean metric, given by

ds^2 = \frac{4 \, |dz|^2}{(1 + |z|^2)^2},

which pulls back the spherical metric under stereographic projection; for elliptic geometry, distances are adjusted to the half-range by taking the minimum between the spherical distance and \pi minus that distance, ensuring geodesics do not exceed \pi/2.^[16]^[15] This model preserves angles exactly due to its conformal nature but distorts distances nonlinearly, with the conformal factor $4 / (1 + |z|^2)^2 scaling lengths more severely farther from the origin, which facilitates the Euclidean-plane drawing of elliptic figures like triangles whose sides appear as circular arcs.^[4]^[14]

Low-Dimensional Cases

Two-dimensional elliptic plane

The two-dimensional elliptic plane is a model of elliptic geometry obtained as the real projective plane \mathbb{RP}^2, or equivalently, as the quotient space of the sphere S^2 by identifying antipodal points.^[1] In this construction, points on the sphere represent lines through the origin in \mathbb{R}^3, and the identification ensures that opposite points are considered the same.^[4] The space is compact and closed, with a finite total area of $2\pi R^2, where R is the radius of the covering sphere.^[1] In the elliptic plane, triangles are formed by arcs of great circles on the sphere, truncated to lengths at most \pi/2 R to avoid redundancy under antipodal identification.^[4] The sum of the interior angles A + B + C of any such triangle exceeds \pi radians, with the angular excess E = A + B + C - \pi serving as a measure of the triangle's size relative to the curvature.^[1] This property arises from the positive Gaussian curvature K = 1/R^2 inherent to the space.^[4] An analog of Girard's theorem holds, stating that the area of a triangle is given by

\text{Area} = R^2 (A + B + C - \pi).

^[17] This formula quantifies how the excess directly corresponds to the enclosed area, scaled by the square of the radius, and applies uniformly to all triangles in the elliptic plane.^[4] A distinctive feature of the elliptic plane is that every pair of "lines"—defined as great circle arcs of length at most \pi/2 R—intersects exactly once, reflecting the absence of parallel lines and the closed topology akin to a projective plane.^[1] This intersection property underscores the finite, bounded nature of the space, where the entire plane can be visualized as a hemisphere with opposite boundary points glued together.^[4]

Three-dimensional elliptic space

Three-dimensional elliptic space, also known as elliptic 3-space, is the real projective space \mathbb{RP}^3, which can be constructed as the 3-sphere S^3 with antipodal points identified, i.e., S^3 / \{\pm 1\}. This identification ensures that every pair of antipodal points on the sphere represents the same point in the projective space, resulting in a compact manifold without boundary that models a closed, finite universe in three dimensions. The metric on elliptic 3-space is the standard round metric induced from the 3-sphere of radius R, where R is the curvature radius, yielding constant positive sectional curvature K = 1/R^2.^[18] The total volume of elliptic 3-space with curvature radius R is \pi^2 R^3. This follows from the fact that the volume of the covering 3-sphere S^3 is $2\pi^2 R^3, and the antipodal quotient map is a 2-to-1 covering, halving the volume.^[19] In elliptic 3-space, planes are embedded elliptic 2-spaces, realized as the quotients of great 2-spheres on S^3 by the antipodal identification. These great 2-spheres are the intersections of S^3 with 3-dimensional linear subspaces through the origin in \mathbb{R}^4, and under the quotient, they become copies of \mathbb{RP}^2. A key feature is that any two such planes intersect, as their preimages on S^3—great 2-spheres—always intersect in a great circle, reflecting the absence of parallel planes in elliptic geometry.^[20] The volume element in elliptic 3-space can be expressed using hyperspherical coordinates inherited from S^3, where the metric is ds^2 = R^2 [d\chi^2 + \sin^2 \chi (d\theta^2 + \sin^2 \theta \, d\phi^2)], with the volume form dV = R^3 \sin^2 \chi \sin \theta \, d\chi \, d\theta \, d\phi. To compute the total volume, integrate over a fundamental domain of the antipodal action, such as \chi \in [0, \pi/2], \theta \in [0, \pi], \phi \in [0, 2\pi], yielding \int_0^{\pi/2} \sin^2 \chi \, d\chi \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi = (\pi/4) \cdot 2 \cdot 2\pi = \pi^2, and thus V = \pi^2 R^3. Alternative parametrizations, such as those using three angles each ranging over [0, \pi/2] with an appropriate Jacobian accounting for the quotient, also integrate to this volume.^[19] In cosmology, elliptic 3-space serves as a model for a closed universe with positive spatial curvature, where the finite volume \pi^2 R^3 implies a compact spatial topology without boundaries or infinite extents. Geodesics in this space are closed loops, meaning there are no infinite rays—all paths eventually return to their starting point, and light signals propagate in finite circuits, potentially leading to observable repeating patterns in the cosmic microwave background if the scale R is sufficiently large.^[21]

Properties and Comparisons

Differences from Euclidean geometry

Elliptic geometry differs fundamentally from Euclidean geometry in its topological structure. While Euclidean space is infinite and non-compact, extending indefinitely in all directions, elliptic space is compact and finite, akin to the surface of a sphere where opposite points are identified, resulting in a closed manifold without boundary.^[1] This compactness implies that any two points can be connected by a unique shortest path of length at most πR, where R is the radius of curvature, and the total "area" or volume is finite.^[4] Metically, elliptic geometry exhibits positive constant curvature, in contrast to the zero curvature of Euclidean geometry. In Euclidean space, distances are measured along straight lines in a flat metric, but in elliptic geometry, the metric is derived from the round metric on the sphere, leading to geodesics that are great circles. Consequently, distances "wrap around" after reaching πR, meaning that traveling far enough along a geodesic returns to the starting point, and the space lacks an infinite extent.^[1]^[22] Axiomatic divergences are most evident in the treatment of parallelism and trigonometry. Playfair's axiom, a formulation of Euclid's parallel postulate stating that through a point not on a given line, exactly one parallel line can be drawn, fails entirely in elliptic geometry, where no parallel lines exist—all lines through a point intersect the given line.^[4] This leads to the intersection of all pairs of lines, generalizing the Pythagorean theorem into spherical trigonometry, where right triangles satisfy relations involving spherical excesses rather than simple proportionality.^[1] A hallmark property of elliptic triangles is that their interior angle sum exceeds π radians (180 degrees), unlike the exact π sum in Euclidean triangles; the excess is proportional to the triangle's area, known as the spherical excess.^[4]^[22] This is reflected in the spherical law of cosines for sides, adapted to elliptic geometry:

\cos\left(\frac{c}{R}\right) = \cos\left(\frac{a}{R}\right) \cos\left(\frac{b}{R}\right) + \sin\left(\frac{a}{R}\right) \sin\left(\frac{b}{R}\right) \cos C

where a, b, c are side lengths opposite angles A, B, C, and R is the curvature radius; this contrasts with the Euclidean law c² = a² + b² - 2ab cos C.^[23]

Self-consistency and parallel postulate

Elliptic geometry demonstrates internal consistency through its embedding in well-established models such as projective spaces and spherical geometries, where all axioms hold without contradiction when appropriately interpreted. In the projective model, lines are defined as intersections of planes through the origin in a higher-dimensional Euclidean space, ensuring incidence relations are preserved and reducing the geometry to analytic computations within Euclidean coordinates. Similarly, the spherical model identifies antipodal points on a unit sphere to eliminate the double intersection issue of great circles, thereby satisfying the axioms of incidence and congruence while avoiding logical inconsistencies.^[24]^[25] The parallel postulate in elliptic geometry is resolved by rejecting the existence of parallel lines altogether: through any point not on a given line, no parallel line exists, as all lines intersect within the finite space. This reinterpretation aligns with Riemann's hypothesis of positive constant curvature, where the geometry's closure ensures universal intersection, contrasting with the Euclidean case of exactly one parallel. The adaptation of Hilbert's axioms for non-Euclidean spaces, particularly by modifying the parallel axiom (Hilbert's Axiom I-7), confirms this structure's coherence without reliance on the Euclidean postulate.^[26]^[4] A sketch of consistency can be seen via the spherical excess formula, where the sum of angles in a triangle exceeds π radians, proportional to the enclosed area on the sphere. Assuming Euclidean parallels in this model would imply infinite extent, contradicting the finite volume and universal intersection of the elliptic space, thus proving no such parallels can exist without violating the model's boundedness. In the 19th century, Carl Friedrich Gauss, Bernhard Riemann, and Felix Klein established this consistency rigorously by reducing elliptic geometry to Euclidean geometry through coordinate systems, such as Klein's use of projective coordinates to embed the elliptic plane in real projective space.^[24]^[26]

Higher Dimensions and Generalizations

Hyperspherical constructions

Hyperspherical constructions generalize the spherical model of elliptic geometry to higher dimensions by forming the elliptic n-manifold as the quotient space S^n / \{\pm 1\}, where S^n is the n-dimensional sphere of radius R equipped with its standard round metric, and the identification is via the antipodal map.^[27] This quotient inherits a Riemannian metric of constant sectional curvature +1/R^2, making it a complete, simply connected space only in the universal cover sense, but compact and homogeneous in the elliptic structure.^[27] Coordinates on this space can be introduced recursively using hyperspherical angles \theta_1, \theta_2, \dots, \theta_n, where each \theta_i parameterizes nested spheres, analogous to latitude and longitude generalizations. The line element takes the form ds^2 = R^2 \left( d\theta_1^2 + \sin^2 \theta_1 \, ds_{n-1}^2 \right) on S^n, with the quotient metric adjusted for the identification, restricting angles appropriately (e.g., \theta_i \in [0, \pi] with antipodal symmetry). The volume form for integration involves products of sine powers, specifically \sin^{n-1} \theta_1 \sin^{n-2} \theta_2 \cdots \sin \theta_{n-1} \, d\theta_1 \wedge \cdots \wedge d\theta_n on the sphere, halved under the quotient to yield the elliptic measure. The total volume of the elliptic n-manifold with this metric is given by

V_n = \frac{\pi^{(n+1)/2}}{\Gamma\left( \frac{n+1}{2} \right)} R^n,

which is half the volume of the covering sphere S^n, reflecting the twofold covering map.^[28] For example, in three dimensions, this yields a volume of \pi^2 R^3, consistent with the elliptic 3-space model. A distinctive topological feature is the orientability of the elliptic n-manifold: it is orientable when n is odd and non-orientable when n is even, mirroring the properties of real projective space \mathbb{RP}^n.^[29] This arises from the action of the antipodal map on the orientation, which reverses it in even dimensions but preserves it in odd dimensions.

Projective extensions

In projective geometry, elliptic spaces emerge as metric realizations of real projective spaces equipped with a constant positive curvature metric derived from a non-degenerate quadratic form. The two-dimensional case identifies the elliptic plane with the real projective plane \mathbb{RP}^2, whose points are the 1-dimensional subspaces of \mathbb{R}^3 (or lines through the origin), and whose lines are the 2-dimensional subspaces (or planes through the origin). This structure is obtained by quotienting the unit 2-sphere S^2 by antipodal identification, yielding a compact surface where every pair of lines intersects exactly once.^[11] The metric on \mathbb{RP}^2 is induced by the round metric on S^2, with the distance d between two points represented by unit vectors \mathbf{u} and \mathbf{v} in \mathbb{R}^3 given by d = \arccos(|\mathbf{u} \cdot \mathbf{v}|), ensuring geodesic lengths range from 0 to \pi/2 and constant Gaussian curvature 1.^[30] Geodesics, or elliptic lines, correspond to great circle arcs on S^2 of length at most \pi, and the total area of the space is $2\pi. This model satisfies the elliptic parallel postulate—no parallels exist—and eliminates the plane separation axiom, rendering the space non-orientable.^[31] Extensions to higher dimensions generalize this construction: the n-dimensional elliptic space is the real projective space \mathbb{RP}^n, comprising 1-dimensional subspaces of \mathbb{R}^{n+1}, with the metric lifted from the standard round metric on the n-sphere S^n via antipodal quotient. Here, points are equivalence classes [\mathbf{x}] for \mathbf{x} \in \mathbb{R}^{n+1} \setminus \{\mathbf{0}\}, and the distance is d([\mathbf{u}], [\mathbf{v}]) = \arccos(|\mathbf{u} \cdot \mathbf{v}|) for unit representatives, yielding constant sectional curvature 1 and compactness. The volume of \mathbb{RP}^n is half that of S^n, specifically \frac{\pi^{(n+1)/2}}{\Gamma\left( \frac{n+1}{2} \right)}.^[32] In this setting, any two geodesics intersect, and the space serves as a model for elliptic geometry in arbitrary dimensions, though orientability holds only for odd n.^[33] This projective framework, pioneered by von Staudt and advanced by Klein in his Erlangen program, treats elliptic geometry as a specialization of projective geometry via an absolute conic or polarity, unifying it with hyperbolic and Euclidean geometries through transformations preserving incidence. Such extensions highlight the projective space's role as an abstract elliptic geometry devoid of metric, to which curvature is added conformally.^[34]

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